Generally, the cops will go perform what's called a "welfare check". If when they arrive at the scene they see any signs that the person has harmed themselves or is incapacitated, they'll summon paramedics to deal with it. Depending on where she lives, the laws could differ pretty dramatically. In a lot of places, hospitals are allowed to hold someone for a 72 hour evaluation period. The person will be seen by psychologists, and if it's determined that they're a threat to themselves, a judge can order them committed involuntarily.

If the cops arrive and nothing has happened, whoever makes the call is NOT going to get in trouble with direct evidence that suicide was threatened, which would be the Plurk or chat log where it took place. It's worth doing if for no other reason than it could give her a serious wake-up call that when you say this kind of shit, it has serious consequences, such as police at your house and the chance that you could wind up committed, which frankly it seems like she needs to be.